Analytical Ramsey Scheme for a single TLS

Here, we derive the exact analytical expression for Ramsey fringes.

General solution of the TLS dynamics

In [1]:
from sympy import Function, symbols, sqrt, Eq, Abs, exp, cos, sin, Matrix, dsolve, solve, S, lambdify
from sympy import I as 𝕚, pi as π
In [2]:
g = Function('g')
e = Function('e')
t, τ = symbols('t, tau', positive=True)
Δ = symbols('Delta', real=True)
η = symbols('eta', positive=True)
Ω = symbols('Omega', real=True)
g0, e0 = symbols('g_0, e_0')
In [3]:
Ĥ = Matrix([
    [-Δ/2,  η ],
    [  η,  Δ/2]
])
Ĥ
Out[3]:
$\displaystyle \left[\begin{matrix}- \frac{\Delta}{2} & \eta\\\eta & \frac{\Delta}{2}\end{matrix}\right]$
In [4]:
TDSE_system = [
    g(t).diff(t) + 𝕚 * (Ĥ[0,0] * g(t) + Ĥ[0,1] * e(t)),
    e(t).diff(t) + 𝕚 * (Ĥ[1,0]*  g(t) + Ĥ[1,1] * e(t)),
]
In [5]:
sols_gen = dsolve(TDSE_system, [g(t), e(t)])
In [6]:
effective_rabi_freq = {
    Ω: sqrt(Δ**2 + (2*η)**2)
}
In [7]:
Eq(Ω, effective_rabi_freq[Ω])
Out[7]:
$\displaystyle \Omega = \sqrt{\Delta^{2} + 4 \eta^{2}}$
In [8]:
find_Ω = {
    sqrt(-Δ**2 - 4 * η**2): 𝕚 * Ω,
    sqrt(Δ**2 + 4 * η**2): Ω
}
In [9]:
sols_gen[0].subs(find_Ω)
Out[9]:
$\displaystyle g{\left(t \right)} = - \frac{C_{1} \left(\Delta + \Omega\right) e^{\frac{i \Omega t}{2}}}{2 \eta} - \frac{C_{2} \left(\Delta - \Omega\right) e^{- \frac{i \Omega t}{2}}}{2 \eta}$
In [10]:
sols_gen[1].subs(find_Ω)
Out[10]:
$\displaystyle e{\left(t \right)} = C_{1} e^{\frac{i \Omega t}{2}} + C_{2} e^{- \frac{i \Omega t}{2}}$

Analytical solution for initial-value Rabi cycling

We specialize the above general solution to an arbitrary initial state with complex amplidues $g_0$, $e_0$.

In [11]:
boundary_conditions = {
    t: 0,
    g(t): g0,
    e(t) : e0,
}
In [12]:
find_4ηsq = {
    Ω**2 - Δ**2: 4 * η**2
}
In [13]:
C1, C2 = symbols("C1, C2")
In [14]:
_integration_constants = solve(
    [sol.subs(find_Ω).subs(boundary_conditions) for sol in sols_gen],
    [C1, C2]
)
integration_constants = {
    k: v.subs(find_4ηsq)
    for (k, v) in _integration_constants.items()
}
In [15]:
Eq(C1, integration_constants[C1])
Out[15]:
$\displaystyle C_{1} = \frac{- \Delta e_{0} + \Omega e_{0} - 2 \eta g_{0}}{2 \Omega}$
In [16]:
Eq(C2, integration_constants[C2])
Out[16]:
$\displaystyle C_{2} = \frac{\Delta e_{0} + \Omega e_{0} + 2 \eta g_{0}}{2 \Omega}$
In [17]:
def simplify_sol_g(sol_g):
    rhs = (
        sol_g
        .rhs
        .rewrite(sin)
        .expand()
        .collect(𝕚)
        .subs(effective_rabi_freq)
        .simplify()
        .subs(find_Ω)
    )
    return Eq(sol_g.lhs, rhs)
In [18]:
sol_g = simplify_sol_g(sols_gen[0].subs(find_Ω).subs(integration_constants))
sol_g
Out[18]:
$\displaystyle g{\left(t \right)} = \frac{i \Delta g_{0} \sin{\left(\frac{\Omega t}{2} \right)} + \Omega g_{0} \cos{\left(\frac{\Omega t}{2} \right)} - 2 i e_{0} \eta \sin{\left(\frac{\Omega t}{2} \right)}}{\Omega}$
In [19]:
sol_e = (
    sols_gen[1]
    .subs(find_Ω)
    .subs(integration_constants)
    .expand()
    .rewrite(sin)
    .expand()
)
sol_e
Out[19]:
$\displaystyle e{\left(t \right)} = - \frac{i \Delta e_{0} \sin{\left(\frac{\Omega t}{2} \right)}}{\Omega} + e_{0} \cos{\left(\frac{\Omega t}{2} \right)} - \frac{2 i \eta g_{0} \sin{\left(\frac{\Omega t}{2} \right)}}{\Omega}$

For example, when starting from the ground state:

In [20]:
sol_g.subs({g0: 1, e0: 0})
Out[20]:
$\displaystyle g{\left(t \right)} = \frac{i \Delta \sin{\left(\frac{\Omega t}{2} \right)} + \Omega \cos{\left(\frac{\Omega t}{2} \right)}}{\Omega}$
In [21]:
sol_e.subs({g0: 1, e0: 0})
Out[21]:
$\displaystyle e{\left(t \right)} = - \frac{2 i \eta \sin{\left(\frac{\Omega t}{2} \right)}}{\Omega}$

For the population dynamics, we find:

In [22]:
def abs_sq(eq):
    lhs = eq.lhs
    rhs = eq.rhs
    return Eq(Abs(lhs)**2, (rhs * rhs.conjugate()).expand())
In [23]:
abs_sq(sol_g.subs({g0: 1, e0: 0}))
Out[23]:
$\displaystyle \left|{g{\left(t \right)}}\right|^{2} = \frac{\Delta^{2} \sin^{2}{\left(\frac{\Omega t}{2} \right)}}{\Omega^{2}} + \cos^{2}{\left(\frac{\Omega t}{2} \right)}$
In [24]:
abs_sq(sol_e.subs({g0: 1, e0: 0}))
Out[24]:
$\displaystyle \left|{e{\left(t \right)}}\right|^{2} = \frac{4 \eta^{2} \sin^{2}{\left(\frac{\Omega t}{2} \right)}}{\Omega^{2}}$

Initial π/2 pulse

In [25]:
T = π / (4*η)

For the ground state amplitude, we find:

In [26]:
sol_g_πhalf = sol_g.subs({g0: 1, e0: 0}).subs({t:T})
sol_g_πhalf
Out[26]:
$\displaystyle g{\left(\frac{\pi}{4 \eta} \right)} = \frac{i \Delta \sin{\left(\frac{\pi \Omega}{8 \eta} \right)} + \Omega \cos{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega}$
In [27]:
sol_g_πhalf.subs({g0: 1, e0: 0}).subs(effective_rabi_freq).subs({Δ: 0, η: 15})
Out[27]:
$\displaystyle g{\left(\frac{\pi}{60} \right)} = \frac{\sqrt{2}}{2}$
In [28]:
sol_g_πhalf.subs({g0: 1, e0: 0}).subs(effective_rabi_freq).subs({Δ: 0, η: 15}).evalf()
Out[28]:
$\displaystyle g{\left(\frac{\pi}{60} \right)} = 0.707106781186548$
In [29]:
sol_g_πhalf.subs({g0: 1, e0: 0}).subs(effective_rabi_freq).subs({Δ: S(1)/5, η: 15}).evalf()
Out[29]:
$\displaystyle g{\left(\frac{\pi}{60} \right)} = 0.70709443987448 + 0.00471402272699148 i$

For the excited state amplitude, we find:

In [30]:
sol_e_πhalf = sol_e.subs({g0: 1, e0: 0}).subs({t:T})
sol_e_πhalf
Out[30]:
$\displaystyle e{\left(\frac{\pi}{4 \eta} \right)} = - \frac{2 i \eta \sin{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega}$
In [31]:
sol_e_πhalf.subs({g0: 1, e0: 0}).subs(effective_rabi_freq).subs({Δ: 0, η: 15})
Out[31]:
$\displaystyle e{\left(\frac{\pi}{60} \right)} = - \frac{\sqrt{2} i}{2}$
In [32]:
sol_e_πhalf.subs({g0: 1, e0: 0}).subs(effective_rabi_freq).subs({Δ: 0, η: 15}).evalf()
Out[32]:
$\displaystyle e{\left(\frac{\pi}{60} \right)} = - 0.707106781186548 i$
In [33]:
sol_e_πhalf.subs({g0: 1, e0: 0}).subs(effective_rabi_freq).subs({Δ: S(1)/5, η: 15}).evalf()
Out[33]:
$\displaystyle e{\left(\frac{\pi}{60} \right)} = - 0.707103409048722 i$

Free time evolution

In [34]:
timeshift_free = {g(τ): g(τ + T), e(τ): e(τ + T)}
In [35]:
sol_g_free = (
    sol_g
    .expand()
    .subs(effective_rabi_freq)
    .subs({η: 0})
    .subs({Δ: symbols("Delta", positive=True)})
    .subs({t:τ})
    .rewrite(exp)
    .expand()
    .subs({symbols("Delta", positive=True): Δ})
    .subs(timeshift_free)
)
sol_g_free
Out[35]:
$\displaystyle g{\left(\tau + \frac{\pi}{4 \eta} \right)} = g_{0} e^{\frac{i \Delta \tau}{2}}$
In [36]:
sol_e_free = (
    sol_e
    .expand()
    .subs(effective_rabi_freq)
    .subs({η: 0})
    .subs({Δ: symbols("Delta", positive=True)})
    .subs({t:τ})
    .rewrite(exp)
    .expand()
    .subs({symbols("Delta", positive=True): Δ})
    .subs(timeshift_free)
)
sol_e_free
Out[36]:
$\displaystyle e{\left(\tau + \frac{\pi}{4 \eta} \right)} = e_{0} e^{- \frac{i \Delta \tau}{2}}$
In [37]:
Ψ_πhalf_ideal = {g0: sqrt(2)/2, e0: -𝕚*sqrt(2)/2}
In [38]:
sol_g_free_ideal = sol_g_free.subs(Ψ_πhalf_ideal)
sol_g_free_ideal
Out[38]:
$\displaystyle g{\left(\tau + \frac{\pi}{4 \eta} \right)} = \frac{\sqrt{2} e^{\frac{i \Delta \tau}{2}}}{2}$
In [39]:
abs_sq(sol_g_free_ideal)
Out[39]:
$\displaystyle \left|{g{\left(\tau + \frac{\pi}{4 \eta} \right)}}\right|^{2} = \frac{1}{2}$
In [40]:
sol_e_free_ideal = sol_e_free.subs(Ψ_πhalf_ideal)
sol_e_free_ideal
Out[40]:
$\displaystyle e{\left(\tau + \frac{\pi}{4 \eta} \right)} = - \frac{\sqrt{2} i e^{- \frac{i \Delta \tau}{2}}}{2}$
In [41]:
abs_sq(sol_e_free_ideal)
Out[41]:
$\displaystyle \left|{e{\left(\tau + \frac{\pi}{4 \eta} \right)}}\right|^{2} = \frac{1}{2}$
In [42]:
Ψ_πhalf_actual = {g0: sol_g_πhalf.rhs, e0: sol_e_πhalf.rhs}
In [43]:
sol_g_free_actual = sol_g_free.subs(Ψ_πhalf_actual).expand()
sol_g_free_actual
Out[43]:
$\displaystyle g{\left(\tau + \frac{\pi}{4 \eta} \right)} = \frac{i \Delta e^{\frac{i \Delta \tau}{2}} \sin{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega} + e^{\frac{i \Delta \tau}{2}} \cos{\left(\frac{\pi \Omega}{8 \eta} \right)}$
In [44]:
sol_e_free_actual = sol_e_free.subs(Ψ_πhalf_actual).expand()
sol_e_free_actual
Out[44]:
$\displaystyle e{\left(\tau + \frac{\pi}{4 \eta} \right)} = - \frac{2 i \eta e^{- \frac{i \Delta \tau}{2}} \sin{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega}$

Final π/2 pulse

In [45]:
Ψ_free_actual = {g0: sol_g_free_actual.rhs, e0: sol_e_free_actual.rhs}
In [46]:
timeshift_final = {g(T): g(2*T + τ), e(T): e(2*T + τ)}
In [47]:
sol_g_final = sol_g.subs(Ψ_free_actual).subs({t:T}).expand().subs(timeshift_final)
sol_g_final
Out[47]:
$\displaystyle g{\left(\tau + \frac{\pi}{2 \eta} \right)} = - \frac{\Delta^{2} e^{\frac{i \Delta \tau}{2}} \sin^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{2}} + \frac{2 i \Delta e^{\frac{i \Delta \tau}{2}} \sin{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega} + e^{\frac{i \Delta \tau}{2}} \cos^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)} - \frac{4 \eta^{2} e^{- \frac{i \Delta \tau}{2}} \sin^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{2}}$
In [48]:
pop_g_final = abs_sq(sol_g_final)
pop_g_final
Out[48]:
$\displaystyle \left|{g{\left(\tau + \frac{\pi}{2 \eta} \right)}}\right|^{2} = \frac{\Delta^{4} \sin^{4}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{4}} + \frac{2 \Delta^{2} \sin^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{2}} + \frac{4 \Delta^{2} \eta^{2} e^{i \Delta \tau} \sin^{4}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{4}} + \frac{4 \Delta^{2} \eta^{2} e^{- i \Delta \tau} \sin^{4}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{4}} - \frac{8 i \Delta \eta^{2} e^{i \Delta \tau} \sin^{3}{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{3}} + \frac{8 i \Delta \eta^{2} e^{- i \Delta \tau} \sin^{3}{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{3}} + \cos^{4}{\left(\frac{\pi \Omega}{8 \eta} \right)} - \frac{4 \eta^{2} e^{i \Delta \tau} \sin^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{2}} - \frac{4 \eta^{2} e^{- i \Delta \tau} \sin^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{2}} + \frac{16 \eta^{4} \sin^{4}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{4}}$
In [49]:
sol_e_final = sol_e.subs(Ψ_free_actual).subs({t:T}).expand().subs(timeshift_final)
sol_e_final
Out[49]:
$\displaystyle e{\left(\tau + \frac{\pi}{2 \eta} \right)} = \frac{2 \Delta \eta e^{\frac{i \Delta \tau}{2}} \sin^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{2}} - \frac{2 \Delta \eta e^{- \frac{i \Delta \tau}{2}} \sin^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{2}} - \frac{2 i \eta e^{\frac{i \Delta \tau}{2}} \sin{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega} - \frac{2 i \eta e^{- \frac{i \Delta \tau}{2}} \sin{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega}$
In [50]:
pop_e_final = abs_sq(sol_e_final)
pop_e_final
Out[50]:
$\displaystyle \left|{e{\left(\tau + \frac{\pi}{2 \eta} \right)}}\right|^{2} = - \frac{4 \Delta^{2} \eta^{2} e^{i \Delta \tau} \sin^{4}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{4}} + \frac{8 \Delta^{2} \eta^{2} \sin^{4}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{4}} - \frac{4 \Delta^{2} \eta^{2} e^{- i \Delta \tau} \sin^{4}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{4}} + \frac{8 i \Delta \eta^{2} e^{i \Delta \tau} \sin^{3}{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{3}} - \frac{8 i \Delta \eta^{2} e^{- i \Delta \tau} \sin^{3}{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{3}} + \frac{4 \eta^{2} e^{i \Delta \tau} \sin^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{2}} + \frac{8 \eta^{2} \sin^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{2}} + \frac{4 \eta^{2} e^{- i \Delta \tau} \sin^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)} \cos^{2}{\left(\frac{\pi \Omega}{8 \eta} \right)}}{\Omega^{2}}$

We can find an "ideal" expression in the limit $\eta \gg \Delta$. That is, $\Omega \approx 2\eta$ and $\Delta/\eta \rightarrow 0$:

In [51]:
pop_g_final_ideal = Eq(
    symbols('p_g'),
    pop_g_final
    .subs({Ω: 2*η})
    .rewrite(sin)
    .expand()
    .subs({Δ/η: 0})
    .rhs
)
pop_g_final_ideal
Out[51]:
$\displaystyle p_{g} = \frac{1}{2} - \frac{\cos{\left(\Delta \tau \right)}}{2}$
In [52]:
pop_e_final_ideal = Eq(
    symbols('p_e'),
    pop_e_final
    .subs({Ω: 2*η})
    .rewrite(sin)
    .expand()
    .subs({Δ/η: 0})
    .rhs
)
pop_e_final_ideal
Out[52]:
$\displaystyle p_{e} = \frac{\cos{\left(\Delta \tau \right)}}{2} + \frac{1}{2}$

Note that his is simply $\cos(\Delta\cdot\tau/2)^2$:

In [53]:
assert (pop_e_final_ideal.rhs - cos(Δ*τ/2)**2).simplify() == S(0)

Ramsey Fringes

The "Ramsey fringes" result from plotting the excitated state popultion for varying detunging $\Delta$ and fixed pulse amplitude $\eta$ and time-of-flight $\tau$.

In [54]:
from matplotlib import pylab as plt
import numpy as np
In [55]:
pop_e_func = lambdify([Δ, η, τ], pop_e_final.subs(effective_rabi_freq).rhs)
In [56]:
pop_e_ideal_func = lambdify([Δ, η, τ], pop_e_final_ideal.subs(effective_rabi_freq).rhs)
In [57]:
def plot_ramsey_fringes(
    Δ_min=-1,
    Δ_max=1,
    η=1,
    τ=10.0,
    N=101,
    ax=None,
    label=None,
    figsize=(10, 6),
    func=pop_e_func,
    legend=None,
):
    if ax is None:
        fig, ax = plt.subplots(figsize=figsize)
    Δ_vals = np.linspace(Δ_min, Δ_max, N)
    p_vals = func(Δ_vals, eta=η, tau=τ).real
    ax.plot(Δ_vals, p_vals, label=label)
    ax.set_xlabel("Δ")
    ax.set_ylabel("pₑ")
    if legend is None:
        legend = label is not None
    if legend:
        ax.legend()
    return ax
In [58]:
ax = plot_ramsey_fringes(Δ_min=-20, Δ_max=20, N=1001, label="Ramsey")
#plot_ramsey_fringes(Δ_min=-20, Δ_max=20, N=1001, label="ideal", ax=ax, func=pop_e_ideal_func)
In [59]:
ax = plot_ramsey_fringes(Δ_min=-5, Δ_max=5, N=1001, η=1, label="Ramsey");
ax = plot_ramsey_fringes(Δ_min=-5, Δ_max=5, N=1001, η=1, label="ideal", ax=ax, func=pop_e_ideal_func);
ax.figure.suptitle("Actual vs ideal fringes for small η");
ax.figure.tight_layout()
In [60]:
ax = plot_ramsey_fringes(Δ_min=-5, Δ_max=5, N=1001, η=10, label="Ramsey");
ax = plot_ramsey_fringes(Δ_min=-5, Δ_max=5, N=1001, η=10, label="ideal", ax=ax, func=pop_e_ideal_func);
ax.figure.suptitle("Actual vs ideal fringes for large η");
ax.figure.tight_layout()
In [61]:
ax = plot_ramsey_fringes(Δ_min=0.9, Δ_max=1.1, N=1001, η=1, label="η=1")
plot_ramsey_fringes(Δ_min=0.9, Δ_max=1.1, N=1001, η=10, label="η=10", ax=ax)
plot_ramsey_fringes(Δ_min=0.9, Δ_max=1.1, N=1001, η=100, label="η=100", ax=ax)
plot_ramsey_fringes(Δ_min=0.9, Δ_max=1.1, N=1001, label="ideal", ax=ax, func=pop_e_ideal_func);
ax.figure.suptitle("Probing around Δ=1");
ax.figure.tight_layout()
In [62]:
ax = plot_ramsey_fringes(Δ_min=-1, Δ_max=1, N=1001, τ=10, label="τ=10");
plot_ramsey_fringes(Δ_min=-1, Δ_max=1, N=1001, τ=1, label="τ=1", ax=ax);
plot_ramsey_fringes(Δ_min=-1, Δ_max=1, N=1001, τ=20, label="τ=20", ax=ax);
ax.figure.suptitle("Varying the time of flight");
ax.figure.tight_layout()

Conclusions:

  • The more $η > Δ$, that is, the faster/stronger the Rabi pulses, the better the fringes are described by the "ideal" expression $\cos(\Delta\cdot\tau/2)^2$
  • The width of the fringes is determined by the time of flight $\tau$